Allen Young has always been proud of his personal investment strategies and has done very well over the past several years. He invests primarily in the stock market. Over the past several months, however, Allen has become very concerned about the stock market as a good investment. In some cases, it would have been better for Allen to have his money in a bank than in the market. During the next year, Allen must decide whether to invest $10,000 in the stock market or in a certificate of deposit (CD) at an
interest rate of 5%. If the market is good, Allen believes that he could get a 15% return by investing in stocks. With a fair market, he expects to get a 6% return. If the market is bad, he will most likely to get a 5% loss.
Allen estimates that the probability of a good market is p and the probability of a fair market is q. He thinks that CD is the best option given the probabilities. Now, Allen is thinking about paying for a stock market newsletter, which could predict very accurately how the market would be. He determines that he would at most pay $500 for the newsletter. Please offer an upper bound and a lower bound for Allen’s estimate of p.
Upper bound: 0.95
Lower bound: 0.05
To find the upper and lower bounds for Allen's estimate of p, we need to consider the maximum and minimum values that he would be willing to pay for the newsletter based on his expected returns.
Upper Bound:
Since Allen is willing to pay at most $500 for the newsletter, we can assume that the value he would get from the newsletter should be greater than or equal to $500. Let's calculate the expected return from investing in the stock market with the newsletter.
Expected Return with the Newsletter:
If the market is good, Allen expects a 15% return. Let's denote this return as Rg.
If the market is fair, Allen expects a 6% return. Let's denote this return as Rf.
If the market is bad, Allen expects a 5% loss. Let's denote this return as Rb.
Expected Return = p * Rg + q * Rf + (1 - p - q) * Rb
To ensure the value from the newsletter is greater than or equal to $500, we set up the following inequality:
p * Rg + q * Rf + (1 - p - q) * Rb >= $500
Substituting the given return values, we have:
p * 15 + q * 6 + (1 - p - q) * (-5) >= $500
15p + 6q - 5 + 5p + 5q >= $500
20p + 11q >= $505
To find the upper bound for p, we need to maximize the expression 20p + 11q under the constraint 20p + 11q >= $505. The upper bound for p occurs when q = 0. Substituting q = 0 into the equation, we have:
20p + 11(0) >= $505
20p >= $505
p >= $505 / 20
p >= $25.25
Therefore, the upper bound for Allen's estimate of p is approximately $25.25.
Lower Bound:
To find the lower bound, we consider the minimum value that Allen would be willing to pay for the newsletter, which is $0. In this case, he would not be willing to pay anything for the newsletter, indicating that he has no confidence in its ability to predict the market accurately. Therefore, the lower bound for Allen's estimate of p is 0.
In summary, the upper bound for Allen's estimate of p is approximately $25.25, and the lower bound is 0.
To determine the upper and lower bounds for Allen's estimate of p, we need to consider the maximum and minimum values that he would be willing to pay for the stock market newsletter.
Let's start with the upper bound. Allen has determined that he would be willing to pay at most $500 for the newsletter. This implies that the expected benefit he would receive from the newsletter should be greater than or equal to $500. The benefit would come from accurately predicting whether the market would be good or fair.
If the market is good, Allen expects a 15% return on his investment in stocks, whereas if the market is fair, he expects a 6% return. Therefore, the difference in expected returns between a good market and a fair market is 15% - 6% = 9%.
To make the newsletter worth $500 or more to Allen, we can equate the expected benefit to $500 and set up the following equation:
(9% * $10,000 * p) + (6% * $10,000 * q) ≥ $500
Simplifying the equation, we have:
0.09 * $10,000 * p + 0.06 * $10,000 * q ≥ $500
0.09p + 0.06q ≥ 0.05
Now, let's consider the lower bound. Since Allen believes that a CD is the best option given the probabilities, this implies that the expected benefit from investing in a CD should be at least as good as the expected benefit from the stock market. The expected benefit from a CD is simply the interest earned, which is 5% of $10,000.
Therefore, the lower bound for Allen's estimate of p can be determined by setting up the following equation:
0.05 * $10,000 * p ≥ $500
0.05p ≥ 0.05
Now, we can solve these equations to find the upper and lower bounds for Allen's estimate of p.
Upper bound:
0.09p + 0.06q ≥ 0.05
Lower bound:
0.05p ≥ 0.05
Simplifying the lower bound equation, we have:
p ≥ 1
Since p represents the probability of a good market, it cannot be greater than 1. Therefore, the lower bound of p is 1.
The upper bound of p will depend on the value of q, which represents the probability of a fair market. Without knowing the specific value or constraints on q, it is not possible to determine a specific upper bound for p. However, we can say that the upper bound of p must be less than or equal to 1 because probabilities cannot exceed 1.
In summary, the lower bound for Allen's estimate of p is 1, and the upper bound depends on the value of q but must be less than or equal to 1.